**1. Introduction**

The theory of magnetohydrodynamics (MHD) is highly appreciated for the industrial purposes. It is based on magnetic properties of electrically conducting liquids. The characteristic of MHD field is to generate currents in moving liquid and produce forces that act upon the liquid flow and reconstruct the magnetic field itself. To modify flow features of heat and mass analysis, the applied magnetic field impacts the deferred nanoparticles and reforms their absorption inside the liquid. This efficient phenomenon was first utilized for astrophysical and geophysical related problems. Recently, heat transportation and MHD flows have played significant roles in agricultural engineering, petroleum industries and medical treatment such as MHD strategy used for reduction of blood during surgeries, magnetic cell separation and treatment of certain arterial diseases. Basically, the MHD parameter is not only working as a significant parameter to control the cooling/heating rate but also to achieve desired quality of product for different flows. Further, MHD can be used in continuous casting of metal processing to suppress instabilities and control flow field. In this context, Hayat et al. [1,2] explored

the MHD flow through moving surfaces and concluded that enhancement in magnetic parameter shows increase in nanoparticles concentration and temperature profiles. Hayat et al. [3,4] numerically studied heat transfer impact on MHD axisymmetric third grade liquid flow. Shafiq et al. [5] presented the study of bioconvective MHD tangent hyperbolic nanoliquid flow with Newtonian heating. Shateyi and Makinde [6] prepared MHD stagnant point flow through a radially stretching convectively heated disk. Hayat et al. [7] investigated the third grade axisymmetric MHD flow over a stretched cylinder and showed that momentum layer thickness and velocity profile are increasing when the curvature parameter increases. Moreover, Shafiq et al. [8] discussed magnetohydrodynamics axisymmetric third grade liquid flow between two porous disks.

The novelty of Marangoni convection is generally the edge dissipative layer between two phase fluid flows such as gas–liquid and liquid–liquid interfaces. It depends upon the variation of surface tension driven by temperature, chemical concentration and applied magnetic field. These gradients can occur only when fluid interfaces contain different fluid properties from each other. Due to the viscosity of interacting liquids, external forces such as gravitational and shear forces come into action. Most researchers have focused their interest on simulating these external forces by utilizing governing equations due to its widespread application in the fields of space processing, industrial manufacturing processes and microgravity science. The significance of Marangoni convective flows in the transportation process of heat and mass into different systems have been thoroughly scrutinized in [9–11]. Kumar et al. [12] discussed Marangoni convective Casson nanoliquid flow in the presence of chemical reaction and uniform heat source/sink and observed that Marangoni parameter showed dominant behavior in terms of velocity as well as temperature fields. Din et al. [13] examined the effect of Marangoni convection on based nanoliquid with thermal radiation and demonstrated that decreasing behavior of velocity profile depends on suction parameter, whereas the temperature distribution and boundary layer thickness increased with an increase in nanofluid volume fraction. Sheikholeslami and Ganji [14] studied the impact of magnetic field on nanoliquid flow by Marangoni convection by Runge–Kutta technique and observed that an increment in heat transfer depended on an increment in solid volume fraction of nanofluid. Hayat et al. [15] investigated the impact of radiation and Joule heating on Marangoni mixed convective flow.

For the last few decades, survey of non-Newtonian fluid flows has been the center of attraction for researchers, engineers and scientists. This is due to the application of non-Newtonian liquid flows in the real world, e.g., in bio-engineering, drilling operations, plastic polymers, paint, optical fibers, coated sheets, cosmetics, salt solutions, food item, etc. The existing problems in nature related with larger diameter and higher shear rates can be solved easily; however, when these flows are related to small diameter with low shear rates, the importance of non-Newtonian fluids (see [16,17]) are non-negligible. The deviation from classical Newton's law of viscosity and flow behavior under shear stress to the non-Newtonian fluids become complex. These flows are challenging task for researchers due to their non-linear rheological behavior. Casson liquid model is one of simplest models of non-Newtonian fluids. The idea of Casson fluid administrated by Casson (see [18]) is to build up the blood flow problems. Due to its rheological properties, Casson liquid behaves as a soft solid when yield stress is higher than shear stress, whereas, if shear stress approaches to infinity, then it starts to deform (see [19]). This structure is widely used for different materials, such as jelly, chocolate, honey, blood, tomato sauce and condensed fruit juices. Charm and Kurland [20] used Casson fluid model and investigated the viscosity of human blood. Bhattacharyya and Hayat [21] analyzed the Casson fluid on MHD boundary layer flow through shrinking sheet. Kumar et al. [22] investigated the viscous dissipation phenomenon in Casson nanoliquid over a moving radiative surface. Moreover, Casson fluid flow model [23–25] has been considered for different geometries and various effects in the literature.

The introduction of nanoparticles in different systems is most favorable to intensify thermal conductivity of classical liquid flows, convection heat transfer coefficient and to control loss in energy. The advantages of nanosize particles in fluid systems is to increase surface area, capacity of heat transfer, intensify the flow interface after collision and interact fluid particles with each other. Thus, this phenomenon is a backbone of the industrial processes and is also beneficial for solar energy resources and bio-medical treatment (see [26–34]). The proficiency of the solar systems [35] can be improved by incorporating the nanoparticles as working fluid into the systems. The iron based nanoparticles may be utilized as drug and radiation transportation for the treatment of cancer patient (see [36,37]). Using magnets the particles can be enter through blood stream to tumor. This type of cancer treatment permits high local doses of drugs into the body without any significant side effect. Further, micelles nanoparticles have been recently introduced to target the kidney cells diseases. These particles can pass into the kidney and remain there. Similarly, magnetic based nanoparticles are also used for cell separation, hyperthermia therapies and for the increment in Magnetic Resonance Imaging (MRI) with contrast behavior. Hayat et al. [38] judged that nanofluid enhanced the temperature and associated boundary layer width of Casson flow. Naseem et al. [39] numerically investigated third grade nanoliquid flow using the Cattaneo–Christov model over a Riga plate and observed that, with an increment in thermal and concentration relaxation parameters, a reduction occurred in concentration and temperature distribution, respectively. Rasool et al. [40] examined the MHD Darcy–Forchheimer nanoliquid flow under the nonlinear stretched surface. Rashid et al. [41] investigated the entropy generation in Darcy–Forchheimer flow of nanofluid with five nanomaterials due to stretching cylinder. Naseem et al. [42] considered the MHD biconvective flow of a Powell–Eyring nanoliquid over a stretching plate. Rasool et al. [43–48] reported some interesting results involving the role of nanoparticles in typical base fluids flowing over different surfaces.

In the studies mentioned above, one can see that an utmost attention is given to natural convection and heat and mass transfer analysis but less importance has been given to the convection through Marangoni phenomena especially in nanofluid flows. The thermo-capillary and soluto-capillary affects are the main factors in Marangoni convection of fluids and nanofluids. Furthermore, flat surfaces with linear stretching are assumed frequently but axisymmetric analyses are less reported. The main contribution of this research is to examine the process of heat and mass transportation for axisymmetric Marangoni convective flow with an inclined MHD by taking Casson nanofluid flowing towards an infinite disk. Brownian motion and thermophoresis are deliberated on account of nanoparticles structure. Finally, the problem is solved by an accurate numerical technique known as Runge–Kutta fourth-order algorithm, whereas previous studies are given mostly by HAM.
